proof that Euler’s constant exists

Theorem 1

The limit

\\gamma=\\lim_{n\\to\\infty}\\left(\\sum_{k=1}^{n}\\frac{1}{k}-\\ln n\\right)

exists.

Proof. Let

C_{n}=\\frac{1}{1}+\\frac{1}{2}+\\cdots+\\frac{1}{n}-\\ln n

and

D_{n}=C_{n}-\\frac{1}{n}

Then

C_{n+1}-C_{n}=\\frac{1}{n+1}-\\ln\\left(1+\\frac{1}{n}\\right)

and

D_{n+1}-D_{n}=\\frac{1}{n}-\\ln\\left(1+\\frac{1}{n}\\right)

Now, by considering the Taylor series for $\\ln(1+x)$ , we see that

\\frac{1}{n+1}<\\ln\\left(1+\\frac{1}{n}\\right)<\\frac{1}{n}

and so

C_{n+1}-C_{n}<0<D_{n+1}-D_{n}

Thus, the $C_{n}$ decrease monotonically, while the $D_{n}$ increase monotonically, since the differences are negative (positive for $D_{n}$ ). Further, $D_{n}<C_{n}$ and thus $D_{1}=0$ is a lower bound for $C_{n}$ . Thus the $C_{n}$ are monotonically decreasing and bounded below, so they must converge.

References

1 E. Artin, The Gamma Function, Holt, Rinehart, Winston 1964.